Let's suppose your domicile remains state #1, and you are temporarily living in state #2.
You will be required to file a non-resident tax return for state #2 that declares all your income earned while working in that state. That will mean you will have allocate your freelance income to state #1 or #2 based on where you were living when you did each project or job.
Then, you will file a resident return for state #1 that reports all your world-wide income. State #1 will give you at least a partial credit for taxes paid to state #2. In practice, you should end up paying no more than whichever state has the higher tax rate, but not be double taxed. In Turbotax, prepare the non-resident state #2 return first, so the credit flows correctly to state #1. Turbotax will ask you to allocate both your graduate student income (if you receive a stipend) and your freelance income.
Now, beware that some states are aggressive about determining residency and will want to treat you as a resident of state #2 if you live in the state more than 183 days of the year. You would file a resident return for both state #1 and state #2 reporting your total world-wide income. There still should be a reciprocal tax credit but it's more complicated. It would help to know which states we are talking about.
